Sliding mode control (SMC) is a relatively new development in control systems which was first developed in Russia in the 1950's. It is a subclass of Variable Structure Control (VSC), control systems that switch between structures in a non-linear manner in order to drive phase states of the system toward a phase plane trajectory. The phase states of the system include the position error and its derivatives (velocity, acceleration, etc.); the location of the phase states within a phase plane define the state of the system at any given time, and the movement of the phase states through the phase plane is referred to as the phase state trajectory.
When the SMC is operating in a first structure, the phase states follow a first phase state trajectory, and when the SMC is operating in a second structure, the phase states follow a second phase state trajectory. By switching between the first and second structures, the phase states are driven toward a third phase state trajectory, referred to as the sliding line (or hperplane for higher order systems), defined within the phase plane where the first and second phase state trajectories intersect in opposite directions. The switching action is controlled by the location of the phase states relative to the sliding line; when the phase states cross the sliding line while following the first phase state trajectory, the system switches to the second structure to drive the phase states toward the sliding line by following the second phase state trajectory. In this manner, the phase states continuously switch across the sliding line as they follow the sliding line toward the origin of the phase plane (i.e., sliding mode). SMC has the advantage in that the closed loop response is defined by parameters in the controller and it is substantially insensitive to parameter variations in the plant and external load disturbances.
Initially, SMC systems were developed and implemented in continuous-time wherein the phase states are continuously monitored such that the system switches structures instantaneously when the phase states cross the sliding line. An inherent problem with this approach is that the continuous switching action may induce undesirable noise in the system (electrical and acoustic) and it may excite modelled (as well as unmodelled) system dynamics. The above referenced co-pending patent application entitled "Improved Chatter Reduction in Sliding Mode Control of a Disk Drive Actuator" discloses a method for reducing the amount of switching noise by defining a boundary layer around the sliding line and switching between structures only when the phase states exceed the boundary layer. Although this technique reduces the switching noise, there are other inherent problems with continuous-time SMC. Namely, to achieve the desired robustness to parameter variations and external disturbances, it can require gains in the individual structures that exceed the control effort limitations (e.g., exceed the available drive current).
Discrete-time SMC is a more recent development which addresses the drawbacks of continuous-time SMC by combining a conventional linear control effort with a discrete-time sliding mode control effort. This approach is discussed by Weibing Gao in "Discrete-Time Variable Structure Control Systems," IEEE Transactions on Industrial Electronics, Vol. 42, No. 2, April 1995. With discrete-time SMC, the phase states are monitored in discrete-time and the system switches between structures at the sampling rate rather than continuously. Thus, there is an inherent boundary layer about the sliding line with a width defined by the sampling period, as well as other parameters of the controller. Rather than switch continuously such that the phase states "slide" along the sliding line, the phase states switch across the sliding line in a zigzag manner while sliding toward the origin of the phase plane. Another characteristic of discrete-time SMC is that nominally the phase states will cross the sliding line at every sampling period, and the absolute magnitude of the distance between the phase states and the sliding line will remain within the inherent boundary layer.
The linear control effort in discrete-time SMC decreases the necessary switching control effort while still maintaining a certain degree of insensitivity to parameter variations and external load disturbances. However, the system's insensitivity to parameter variations depends on the width of the boundary layer; the system's robustness can be increased by increasing the boundary layer, but this requires an increase in the switching control effort and eventually leads to the same problems associated with continuous-time SMC. Increasing the boundary layer also limits the maximum slope of the sliding line since the sliding line must be constrained within a particular region of the phase plane in order to maintain global stability.
There is, therefore, the need for an improved discrete-time sliding mode controller that exhibits the desired insensitivity to parameter variations and external load disturbances, without increasing the switching noise and without requiring a large control effort in the switching term. A further object of the present invention is to provide a discrete-time sliding mode controller wherein the slope of the sliding line can be increased to a more optimal value within the phase plane while maintaining global stability.